How do I use loophole for Scholarships to get best AOC?

Our Dependent Full-Time College Student received a 1098T.
Box 1 $10,075.90
Box 5 $16,500

The Room and Board for 2020 was $8735 according to the college's statements for Spring and Fall, so I think we should be able to use at least that amount to shift some of the qualified expenses to our child's tax return in order for us to claim the American Opportunity Credit.

However, that is not how the 1098-T shows the charges and payments. They do not include the Spring 2020 charges or our personal check sent in late Dec 2019 for 2020 because they apparently included it in the 2019 1098-T (which I did not realize until just now while inspecting this 1098-T for a very long time trying to understand it.)
And, the school included the Spring 2021 qualified charges on this 2020 1098-T.
But, they included the scholarships for Spring 2020 and Fall 2020 and the amount we paid for Fall 2020 and Spring 2021.
Does it matter which year you pay the costs in? 
How would I calculate this mess of different years in order to qualify for the AOC?

I figured it doesn't matter when we paid when considering the scholarship money, so I am thinking that we can use the $8735 room and board listed on the statements for Spring and Fall 2020 as opposed to what is on the 1098-T because room and board is not included on the 1098-T, only the qualified expenses are on there.
Is that okay?

And, I'm wondering if I can shift even more expenses than room and board to get a bigger credit. Do we have to prove other expenses in some way? What can we count as non-qualified expenses to help us increase the AOC?

Also, when I was testing this, I entered a round $10,000 in my child's tax return as scholarship income. But, the dependent child's standard deduction seems to cover it entirely and their return shows they owe no taxes at all. Can that be correct?

Meanwhile, our parent return went from $0 AOC to $2434 for AOC after testing it with $10,000 instead of the just the 2020 room and board of $8735. I know the maximum AOC is $2500, so I'm wondering if we are allowed to count more than the room and board, and, if so, what else we can count?

Also, 2020 will be the 3rd tax year for trying to qualify for AOC. So, we will have either 2021 or 2022 to qualify again. I expect our child's scholarship to increase by $1,000 in the 2021-2022 school year, but the room and board will decrease because they are moving off campus and we think it will be significantly cheaper. How can we decide which year would be better to try to get the AOC?

Also, I thought we could get only either the AOC or the LLC. But, I saw another post where the expert was saying you could do 4 years of AOC and use the LLC in the year of graduation or flip-flop it. Is this true? 

One other thing about entry of this loophole: In our parent return, I entered the 1098-T information as it is on the form. Then, when I got to the place in the interview where it asked, "Did Child Pay for Room and Board with a Scholarship or Grant?" I answered yes, and entered the temporary $10,000 in the box that said Total used. That seemed to work and flow to Form 8863 properly and then to Schedule 3 and the 1040. Does that all seem correct?

In my child's return, I went to the Education Deduction, Clicked on Expenses and Scholarships, entered the 1098-T as it is on the form, clicked on Scholarships/Grans Edit, then was confused because it asked if some of the scholarship was not for 2020 expenses. But, I didn't notice the last few years, so I'm not sure what to do now. I said yes, and entered $8,500.

Then, it asked if we paid room and board with the scholarship. I answered yes, and said that out of the $16,500 my child received in scholarships , $10,000 was used to pay for room, board, or equipment, etc expenses. Is this all okay?

If that is all the correct method to enter the loophole changes correctly, I just need to understand the specifics of which year's room and board to enter and what other expenses I can count. I know my child bought some books and a computer (or two? - they keeps trying to upgrade on the cheap). But, I know the computer was bought second-hand with cash off Facebook Marketplace - no receipt. Can that be counted? One was also stolen, but I'm sure that doesn't matter here.

And, finally, if I've messed up the scholarships against the expenses, do I have to go back and amend my other tax returns? This is all such a nightmare!!!